Question

theorem of Angle bisector??

Answer 1

Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D between B and C. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC:

and conversely, if a point D on the side BC of triangle ABC divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle ∠ A.

The generalized angle bisector theorem states that if D lies on the line BC, then

This reduces to the previous version if AD is the bisector of ∠ BAC. When D is external to the segment BC, directed line segments and directed angles must be used in the calculation.

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Answer 2

hello ☺

Given : 

A ΔABC, in which AD is the bisector of the exterior ∠A and intersects BC produced in D.

Prove that :
 BD ➗CD = AB ➗ AC 

Construction : 
Draw CE || DA meeting AB in E.

  proof :

Statements Reasons
CE || DA

By construction
∠1 = ∠3

Alternate interior angle
∠2 = ∠4

Corresponding angle

CE ||DA and BK is a transversal

AD is a bisector of ∠A
∠1 = ∠2
∠3 = ∠4
AE = AC

If angles are equal then side opposite to them are also equal
BD ➗ CD = BA➗EA

By Basic proportionality theorem
(EC ||AD)

BD ➗CD = AB➗AE
BA = AB and EA = AE
BD➗CD = AB ➗AC
AE = EC

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